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Properties

Label 819.w
Modulus $819$
Conductor $39$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,0,1])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(8,819)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(819\)
Conductor: \(39\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 39.f
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.19773.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(16\) \(17\) \(19\) \(20\)
\(\chi_{819}(8,\cdot)\) \(1\) \(1\) \(-i\) \(-1\) \(-i\) \(i\) \(-1\) \(i\) \(1\) \(1\) \(i\) \(i\)
\(\chi_{819}(512,\cdot)\) \(1\) \(1\) \(i\) \(-1\) \(i\) \(-i\) \(-1\) \(-i\) \(1\) \(1\) \(-i\) \(-i\)